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Suppose $x_i, y_i \in \mathbb{R}^n$, and $F,G: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$. $$F(x, y) = \frac{x\circ Ay}{x^TAy}$$ $$G(x, y) = \frac{x\circ By}{x^TBy}$$ where $A$ and $B$ are some matrices, $\circ$ is component-wise product (Hadamard product). My question is how to build (find) functions $F^{1/2}$ and $G^{1/2}$ such that given $x_2=F(x_0, y_0)$, and $y_2 = G(y_0, x_0)$: $$x_1 = F^{1/2}(x_0, y_0), y_1=G^{1/2}(y_0, x_0)$$ $$x_2 = F^{1/2}(x_1, y_1), y_2 = G^{1/2}(y_1, x_1)$$

Any thoughts are appreciated.

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Maybe you can first think the case n=1 –  bo.gu Mar 16 '13 at 7:35
    
I cannot edit your question but there's a typo [Hadamar]-->[Hadamard] after the name of the mathematician Jacques Hadamard. –  Duchamp Gérard H. E. Mar 16 '13 at 8:04
    
if n=1 then F and G maps everything nonzero to one –  ashim Mar 16 '13 at 15:25
1  
This was also asked at the same time on math.stackexchange: math.stackexchange.com/questions/331802/…. –  Ryan Reich Mar 16 '13 at 15:43
    
It seems to me that bo.gu's "answer" is a sensible suggestion. (It would have been better posted as a comment, but it looks like bo.gu may not have enough reputation points to do so.) Looking at $n=1$ cuts through the clutter and boils the OP's question down to the following: "Suppose $x_i$ and $y_i$ are real numbers. Are there functions $f$ and $g$ such that given $x_2=1$ and $y_2=1$, we have $x_1=f(x_0,y_0)$, $y_1=g(y_0,x_0)$, $x_2=f(x_1,y_1)$, and $y_2=g(y_1,x_1)$." It's not clear (to me, at least) what this means. –  Barry Cipra Mar 16 '13 at 16:51
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